Identifier
-
Mp00146:
Dyck paths
—to tunnel matching⟶
Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00144: Perfect matchings —rotation⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤ
Values
[1,0] => [(1,2)] => [(1,2)] => [(1,2)] => 1
[1,0,1,0] => [(1,2),(3,4)] => [(1,2),(3,4)] => [(1,4),(2,3)] => 2
[1,1,0,0] => [(1,4),(2,3)] => [(1,3),(2,4)] => [(1,3),(2,4)] => 2
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [(1,2),(3,4),(5,6)] => [(1,6),(2,3),(4,5)] => 2
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [(1,2),(3,5),(4,6)] => [(1,5),(2,3),(4,6)] => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [(1,3),(2,4),(5,6)] => [(1,6),(2,4),(3,5)] => 3
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [(1,3),(2,5),(4,6)] => [(1,5),(2,4),(3,6)] => 3
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [(1,4),(2,5),(3,6)] => [(1,4),(2,5),(3,6)] => 3
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [(1,2),(3,4),(5,6),(7,8)] => [(1,8),(2,3),(4,5),(6,7)] => 2
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [(1,2),(3,4),(5,7),(6,8)] => [(1,7),(2,3),(4,5),(6,8)] => 2
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [(1,2),(3,5),(4,6),(7,8)] => [(1,8),(2,3),(4,6),(5,7)] => 3
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [(1,2),(3,5),(4,7),(6,8)] => [(1,7),(2,3),(4,6),(5,8)] => 3
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [(1,2),(3,6),(4,7),(5,8)] => [(1,6),(2,3),(4,7),(5,8)] => 3
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => [(1,3),(2,4),(5,6),(7,8)] => [(1,8),(2,4),(3,5),(6,7)] => 3
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [(1,3),(2,4),(5,7),(6,8)] => [(1,7),(2,4),(3,5),(6,8)] => 3
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => [(1,3),(2,5),(4,6),(7,8)] => [(1,8),(2,4),(3,6),(5,7)] => 3
[1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => [(1,3),(2,5),(4,7),(6,8)] => [(1,7),(2,4),(3,6),(5,8)] => 3
[1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => [(1,3),(2,6),(4,7),(5,8)] => [(1,6),(2,4),(3,7),(5,8)] => 3
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [(1,4),(2,5),(3,6),(7,8)] => [(1,8),(2,5),(3,6),(4,7)] => 4
[1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => [(1,4),(2,5),(3,7),(6,8)] => [(1,7),(2,5),(3,6),(4,8)] => 4
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => [(1,4),(2,6),(3,7),(5,8)] => [(1,6),(2,5),(3,7),(4,8)] => 4
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [(1,5),(2,6),(3,7),(4,8)] => [(1,5),(2,6),(3,7),(4,8)] => 4
[1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => 2
[1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => [(1,2),(3,4),(5,6),(7,9),(8,10)] => [(1,9),(2,3),(4,5),(6,7),(8,10)] => 2
[1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10)] => [(1,2),(3,4),(5,7),(6,8),(9,10)] => [(1,10),(2,3),(4,5),(6,8),(7,9)] => 3
[1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => [(1,2),(3,4),(5,7),(6,9),(8,10)] => [(1,9),(2,3),(4,5),(6,8),(7,10)] => 3
[1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => [(1,2),(3,4),(5,8),(6,9),(7,10)] => [(1,8),(2,3),(4,5),(6,9),(7,10)] => 3
[1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => [(1,2),(3,5),(4,6),(7,8),(9,10)] => [(1,10),(2,3),(4,6),(5,7),(8,9)] => 3
[1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => [(1,2),(3,5),(4,6),(7,9),(8,10)] => [(1,9),(2,3),(4,6),(5,7),(8,10)] => 3
[1,0,1,1,0,1,0,0,1,0] => [(1,2),(3,8),(4,5),(6,7),(9,10)] => [(1,2),(3,5),(4,7),(6,8),(9,10)] => [(1,10),(2,3),(4,6),(5,8),(7,9)] => 3
[1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => [(1,2),(3,5),(4,7),(6,9),(8,10)] => [(1,9),(2,3),(4,6),(5,8),(7,10)] => 3
[1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => [(1,2),(3,5),(4,8),(6,9),(7,10)] => [(1,8),(2,3),(4,6),(5,9),(7,10)] => 3
[1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => [(1,2),(3,6),(4,7),(5,8),(9,10)] => [(1,10),(2,3),(4,7),(5,8),(6,9)] => 4
[1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => [(1,2),(3,6),(4,7),(5,9),(8,10)] => [(1,9),(2,3),(4,7),(5,8),(6,10)] => 4
[1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => [(1,2),(3,6),(4,8),(5,9),(7,10)] => [(1,8),(2,3),(4,7),(5,9),(6,10)] => 4
[1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => [(1,2),(3,7),(4,8),(5,9),(6,10)] => [(1,7),(2,3),(4,8),(5,9),(6,10)] => 4
[1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => [(1,3),(2,4),(5,6),(7,8),(9,10)] => [(1,10),(2,4),(3,5),(6,7),(8,9)] => 3
[1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => [(1,3),(2,4),(5,6),(7,9),(8,10)] => [(1,9),(2,4),(3,5),(6,7),(8,10)] => 3
[1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => [(1,3),(2,4),(5,7),(6,8),(9,10)] => [(1,10),(2,4),(3,5),(6,8),(7,9)] => 3
[1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => [(1,3),(2,4),(5,7),(6,9),(8,10)] => [(1,9),(2,4),(3,5),(6,8),(7,10)] => 3
[1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => [(1,3),(2,4),(5,8),(6,9),(7,10)] => [(1,8),(2,4),(3,5),(6,9),(7,10)] => 3
[1,1,0,1,0,0,1,0,1,0] => [(1,6),(2,3),(4,5),(7,8),(9,10)] => [(1,3),(2,5),(4,6),(7,8),(9,10)] => [(1,10),(2,4),(3,6),(5,7),(8,9)] => 3
[1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => [(1,3),(2,5),(4,6),(7,9),(8,10)] => [(1,9),(2,4),(3,6),(5,7),(8,10)] => 3
[1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => [(1,3),(2,5),(4,7),(6,8),(9,10)] => [(1,10),(2,4),(3,6),(5,8),(7,9)] => 3
[1,1,0,1,0,1,0,1,0,0] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => [(1,3),(2,5),(4,7),(6,9),(8,10)] => [(1,9),(2,4),(3,6),(5,8),(7,10)] => 3
[1,1,0,1,0,1,1,0,0,0] => [(1,10),(2,3),(4,5),(6,9),(7,8)] => [(1,3),(2,5),(4,8),(6,9),(7,10)] => [(1,8),(2,4),(3,6),(5,9),(7,10)] => 3
[1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => [(1,3),(2,6),(4,7),(5,8),(9,10)] => [(1,10),(2,4),(3,7),(5,8),(6,9)] => 4
[1,1,0,1,1,0,0,1,0,0] => [(1,10),(2,3),(4,7),(5,6),(8,9)] => [(1,3),(2,6),(4,7),(5,9),(8,10)] => [(1,9),(2,4),(3,7),(5,8),(6,10)] => 4
[1,1,0,1,1,0,1,0,0,0] => [(1,10),(2,3),(4,9),(5,6),(7,8)] => [(1,3),(2,6),(4,8),(5,9),(7,10)] => [(1,8),(2,4),(3,7),(5,9),(6,10)] => 4
[1,1,0,1,1,1,0,0,0,0] => [(1,10),(2,3),(4,9),(5,8),(6,7)] => [(1,3),(2,7),(4,8),(5,9),(6,10)] => [(1,7),(2,4),(3,8),(5,9),(6,10)] => 4
[1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => [(1,4),(2,5),(3,6),(7,8),(9,10)] => [(1,10),(2,5),(3,6),(4,7),(8,9)] => 4
[1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => [(1,4),(2,5),(3,6),(7,9),(8,10)] => [(1,9),(2,5),(3,6),(4,7),(8,10)] => 4
[1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => [(1,4),(2,5),(3,7),(6,8),(9,10)] => [(1,10),(2,5),(3,6),(4,8),(7,9)] => 4
[1,1,1,0,0,1,0,1,0,0] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => [(1,4),(2,5),(3,7),(6,9),(8,10)] => [(1,9),(2,5),(3,6),(4,8),(7,10)] => 4
[1,1,1,0,0,1,1,0,0,0] => [(1,10),(2,5),(3,4),(6,9),(7,8)] => [(1,4),(2,5),(3,8),(6,9),(7,10)] => [(1,8),(2,5),(3,6),(4,9),(7,10)] => 4
[1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => [(1,4),(2,6),(3,7),(5,8),(9,10)] => [(1,10),(2,5),(3,7),(4,8),(6,9)] => 4
[1,1,1,0,1,0,0,1,0,0] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => [(1,4),(2,6),(3,7),(5,9),(8,10)] => [(1,9),(2,5),(3,7),(4,8),(6,10)] => 4
[1,1,1,0,1,0,1,0,0,0] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => [(1,4),(2,6),(3,8),(5,9),(7,10)] => [(1,8),(2,5),(3,7),(4,9),(6,10)] => 4
[1,1,1,0,1,1,0,0,0,0] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => [(1,4),(2,7),(3,8),(5,9),(6,10)] => [(1,7),(2,5),(3,8),(4,9),(6,10)] => 4
[1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => [(1,5),(2,6),(3,7),(4,8),(9,10)] => [(1,10),(2,6),(3,7),(4,8),(5,9)] => 5
[1,1,1,1,0,0,0,1,0,0] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => [(1,5),(2,6),(3,7),(4,9),(8,10)] => [(1,9),(2,6),(3,7),(4,8),(5,10)] => 5
[1,1,1,1,0,0,1,0,0,0] => [(1,10),(2,9),(3,6),(4,5),(7,8)] => [(1,5),(2,6),(3,8),(4,9),(7,10)] => [(1,8),(2,6),(3,7),(4,9),(5,10)] => 5
[1,1,1,1,0,1,0,0,0,0] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => [(1,5),(2,7),(3,8),(4,9),(6,10)] => [(1,7),(2,6),(3,8),(4,9),(5,10)] => 5
[1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [(1,6),(2,7),(3,8),(4,9),(5,10)] => [(1,6),(2,7),(3,8),(4,9),(5,10)] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)] => [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)] => 6
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Description
The size of the largest partition in the oscillating tableau corresponding to the perfect matching.
Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.
Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.
Map
rotation
Description
The rotation of a perfect matching.
This returns the perfect matching obtained by relabelling $i$ to $i+1$ cyclically.
This returns the perfect matching obtained by relabelling $i$ to $i+1$ cyclically.
Map
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.
This yields the perfect matching with the number of nestings and crossings exchanged.
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
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